Let ZFC set theory, what is the domain of quantification of a formula like $\forall x\phi(x)$? If the domain is the whole Von Neumann Hierarchy $V$ why it is not a problem that it doesn't form a set?
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Asaf Karagila
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lalessandro
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Why should it need to? – Malice Vidrine Sep 10 '14 at 17:56
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1In a theory with only one "sort" of objects (like ZFC and PA) the domain of quantification is "made of" those thing we intend that the tehory is "speaking of" : sets in ZFC and numbers in PA. We can have set theories with urelemets or with classes, in which cases we need two sorts of variables or suitable predicates "restricting" the quantifiers, like $U(x)$ which is true for evrey object in the domain which is an urelement. – Mauro ALLEGRANZA Sep 10 '14 at 19:28
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(I premise that I have a background in philosophy) I don't fully understand the semantics of such expressions. I take for granted that if I have a formula I need a model to refer to in order to state its truth value and that a model is a structure made up of a set $D$ and a function $F$ which assign to each individual constant a member of $D$,to each one-place predicate a subset of $D$,...,to each n-place predicate a subset of $D X D X D X D...n $. – lalessandro Sep 10 '14 at 19:36
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As far as I understand set theory formulas refers to all the sets(because set theory wants to study sets as a whole) and that in ZFC a standard model is the Von Neumann Hierarchy. My question is how can $V$ be taken as a set if it is not the case? How can be evaluated set theory formulas situation? – lalessandro Sep 10 '14 at 19:36
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1You can find many fine and interesting remarks about this issue in Kenneth Kunen, The Foundations of Mathematics (2009). You can see also in Heinz-Dieter Ebbinghaus & Jörg Flum & Wolfgang Thomas, Mathematical logic (2nd ed 1984) : Sec.VII.4,pag.111. – Mauro ALLEGRANZA Sep 11 '14 at 08:34
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@alessandro: Yes! Shouldn't there be a class of mathematical objects $\mathcal M$ to refer to? Defined by some paradigms together with some variant of Leibniz' law of identity? – Lehs Sep 12 '14 at 16:42
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Set theory doesn't happen in vacuum. There's still first-order logic in the meta-level (which is often either some set theory, or a weak number theoretic theory; depending on the philosophical bent of the mathematician).
The quantifiers are objects of the meta-theory, not of $\sf ZFC$. We define their meaning from outside of set theory.
What might be confusing is the fact that $\sf ZFC$ can "internalize" first-order logic, and reinterpret it as sets and define what is a structure and so one and so forth. In which case, a universal quantifier is defined as a set and is interpreted only on a given structure.
But the quantifiers in the axioms of $\sf ZFC$, or generally in the language of set theory, are not internal to the universe of set theory, but rather external and live in a larger universe (in case the meta-theory is a set theory), or they are syntactic objects (in case the meta-theory is a number theory). Those are two different "planes of existence".
Asaf Karagila
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I don't understand how the set-theoretic universe in which the quantifiers are defined is different from the set-theoretic universe set theory is up to define. Assuming that they can't be the same on pain of circularity. – lalessandro Sep 12 '14 at 10:31
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Are you a Platonist? Do you think that there is really just one concrete universe of set theory which exists in a semi-physical sense? If not, then the universe of set theory is either a set in a larger universe of set theory, and the quantification is done in the larger universe; or the universe of set theory is really an imaginative interpretation of a universe, but we really work with syntactic formula and just prove things from a list of axioms, and that we do in first-order logic interpreted in $\sf PRA$ or something similar, there $\forall x$ is not quantified over objects, [...] – Asaf Karagila Sep 12 '14 at 10:34
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[...] but rather it is a syntactical object which you can then instantiate, or generalize or whatever. But there are no "objects" to quantify over. Because we only have strings to push around and manipulate according to some basic rules of our proof system. In case that you are a Platonist, then you can think about this as something which is not quantified over the universe, but again syntactically within the universe. But then, since we prove that $\sf ZFC$ proves $\forall x\varphi$, it must hold in the universe, so for every set in the universe $\varphi$ is true. – Asaf Karagila Sep 12 '14 at 10:37
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I would leave philosophical concerns aside for this matter, I'm interested in the problem from a mathematical point of view. I'm wondering whether a semantic for set theoretic formulas works differently from those of other fields and if so, whether the difference depends on the fact that the domain of quantification is a class instead of a set. I don't understand then when you say 'the universe of set theory is either a set in a larger universe of set theory, and the quantification is done in the larger universe', how can you say that the set theory universe is a set? – lalessandro Sep 12 '14 at 10:59
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Fine. Mathematically how do you work in set theory? Mathematical theories don't live in vacuum. What is your meta-theory? The answer for how to quantify over all sets lies in your answer there. And I say that the set theoretical universe can be a set in a larger universe. All this really points to the fact that you're asking a very good question, but your ability to deal with its answer is very limited. Perhaps reading through a book about logic and set theory could help. (I recommend Kunen "Set Theory" in this aspect, but also Halbeisen's "Combinatorial Set Theory" looks good.) – Asaf Karagila Sep 12 '14 at 11:07
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Let me add, maybe, that "What is your meta-theory" is a perfectly reasonable mathematical question; "What is the correct meta-theory" is a philosophical question; and "What is your preferred meta-theory?" is a personal question. So by talking about meta-theories here and larger universes I am not trying to be philosophical, I'm trying to cover the possible angles for answers for the first question. If, however, you have an understanding of what is the difference between meta-theory and theory, you will often have answers for the other two questions. – Asaf Karagila Sep 12 '14 at 11:13
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I understand what you say when you talk about a meta-theory and a larger universe containing the set-theoretic universe. what I don't understand is what are there inside this larger universe. and what kind of mathematical object it can be, assuming that it can't be a set because, as far as I understood, no set can be equal or greater than V. – lalessandro Sep 12 '14 at 14:19
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More sets? Can you imagine two groups, one is a subgroup of the other? Can you imagine how the rational and real numbers are two ordered sets? Universe is just working internally to a model. This model can be a set in a larger universe, which itself is a set in a larger universe, and so on and so forth. – Asaf Karagila Sep 12 '14 at 14:22
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Ins't the cardinality of the class of ordinals a bound on size to this process? – lalessandro Sep 12 '14 at 14:26
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Classes are definable collections. Working internally to a model of set theory (so the model is the universe), those are not sets; working externally where that model is a set, those classes are sets of the larger universe. Do you think that a countable model of $\sf ZFC$ is a self-contradictory idea? Please read a bit about Skolem's paradox. – Asaf Karagila Sep 12 '14 at 15:28
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You can see Kenneth Kunen, The Foundations of Mathematics (2009), page 16 :
[The context is an informal discussion of ] Axiom 1. Extensionality :
$\forall x y [\forall z(z \in x \leftrightarrow z \in y) \rightarrow x = y]$.
This says that a set is determined by its members, so that if $x,y$ are two sets with exactly the same members, then $x, y$ are the same set. Extensionality also says something about our intended domain of discourse, or universe, which is usually called $V$. Everything in our universe must be a set, since if we allowed objects $x, y$ which aren't sets, such as a duck ($D$) and a pig ($P$), then they would have no members, so that we would have
$\forall z[z \in P \leftrightarrow z \in D \leftrightarrow z \in \emptyset \leftrightarrow FALSE]$,
whereas $P, D, \emptyset$ are all different objects. So, physical objects, such as $P, D$, are not part of our universe.
Now, informally, one often thinks of sets or collections of physical objects, such as $\{ P, D \}$, or a set of ducks, or the set of animals in a zoo. However, these sets are also not in our mathematical universe. Recall that in writing logical expressions, it is understood that the variables range only over our universe, so that a statement such as "$\forall z \ldots $" is an abbreviation for "for all $z$ in our universe $\ldots$". So, if we allowed $\{ P \}$ and $\{ D \}$ into our universe, then $\forall z(z \in {P} \leftrightarrow z \in {D})$ would be true (since $P, D$ are not in our universe), whereas $\{ P \} \ne \{ D \}$.
More generally, if $x, y$ are (sets) in our universe, then all their elements are also in our universe, so that the hypothesis "$\forall z (z \in x \leftrightarrow z \in y)$" really means that $x, y$ are sets with exactly the same members, so that Extensionality is justified in concluding that $x = y$. So, if $x$ is in our universe, then $x$ must not only be a set, but all elements of $x$, all elements of elements of $x$, etc. must be sets.
See also, after the discussion of Russell's Paradox [page 18] :
Theorem 1.6.6. There is no universal set: $\forall z \exists R[R \notin z]$.
So, although we talk informally about the universe, $V$, we see that there really is no such object.
If we want to study models of $\mathsf {ZFC}$, we have to do it into a theory $\mathsf {ZFC}^+$ "stronger" than $\mathsf {ZFC}$, i.e. into a theory capable of proving the existence of a set $Z$ [which is an object of the universe of $\mathsf {ZFC}^+$] "large enough" to act as $V$ for $\mathsf {ZFC}$, i.e. a set of $\mathsf {ZFC}^+$ containing all the objects necessary to satisy the axioms of $\mathsf {ZFC}$.
Mauro ALLEGRANZA
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